This modern and comprehensive guide to long-range dependence and self-similarity starts with rigorous coverage of the basics, then moves on to cover more specialized, up-to-date topics central to current research. These topics concern, but are not limited to, physical models that give rise to...

The Rosenblatt distribution appears as limit in non-central limit theorems. The generalized Rosenblatt distribution is obtained by allowing different power exponents in the kernel that defines the usual Rosenblatt distribution. We derive an explicit formula for its third moment, correcting the...

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0i1,…,ik∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅)...

Methods for parameter estimation in the presence of long-range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long-range dependence in the case of...

Semi-parametric estimation methods of the long-memory exponent of a time series have been studied in several papers, some applied, others theoretical, some using Fourier methods, others using a wavelet-based technique. In this paper, we compare the Fourier and wavelet approaches to the local...

This paper studies the sample path properties of stochastic processes represented by multiple symmetric [alpha]-stable integrals. It relates the "smoothness" of the sample paths to the "smoothness" of the (non-random) integrand. It also contains results about the behavior of the distribution of...

We show that a non-trivial continuous-time strictly [alpha]-stable, [alpha][set membership, variant](0,2), stationary process cannot be represented in distribution as a discrete linear processwhere is a collection of deterministic functions and are independent strictly [alpha]-stable random...

Let X1, X2,..., Xn be jointly [alpha]-stable random variables, 0 < [alpha] < 2, and let p1, p2,..., pn be non-negative numbers. We give a necessary and sufficient condition for E X1 p1 ... Xn pn to be finite.

Let (Xi)[infinity]i = 1 be a stationary, mean-zero Gaussian process with covariances r(k) = EXk + 1 X1 satisfying r(0) = 1 and r(k) = k-DL(k). Consider the two-parameter empirical process for G(Xi), where G is any measurable function. The Functional Law of the Iterated Logarithm as well as a...